An Exploration of Degeneracy in Abelian Varieties of Fermat Type

TL;DR

This paper investigates the degeneracy phenomena in abelian varieties of Fermat type, analyzing their Hodge structures and symmetry groups, and demonstrates that Jacobians of certain hyperelliptic curves are degenerate when the defining integer is odd and composite.

Contribution

It provides a detailed examination of degeneracy in Fermat-type abelian varieties, establishing conditions under which Jacobians are degenerate and exploring various examples.

Findings

Jacobian of hyperelliptic curves y^2=x^m-1 is degenerate when m is odd and composite

Degeneracy affects the structure via Hodge rings and symmetry groups

Multiple phenomena of degeneracy are illustrated through examples

Abstract

The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $m$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.